Question

apply the properties of integer exponent to generate equivalent expressions to (7^3)^7 times 7^20 divided by 6^-2 with only positive exponents, then solve the expression. provide the answer as an improper fraction

Answers

Answered by GPT 3.5
To apply the properties of integer exponents, we need to simplify each term individually and then combine them.
Let's break down each term and simplify:

First, simplify (7^3)^7 as 7^(3*7) = 7^21.
Next, simplify 7^20 as (7^2)^10 = 49^10.
Lastly, simplify 6^-2 as 1/(6^2) = 1/36.

Now, combine the simplified terms:
(7^21)*(49^10)/(1/36)

To multiply the terms, add the exponents of like bases:
7^21 * (7^2)^10 * 49^10 / (1/36)
= 7^21 * 7^20 * 49^10 / (1/36)
= 7^(21+20) * 49^10 / (1/36)
= 7^41 * 49^10 / (1/36).

To divide, subtract the exponents of like bases:
7^41 * 49^10 / (1/36)
= (7^41 * 49^10) * 36.

Now, let's evaluate this expression:
7^41 * 49^10 * 36
= (2^3)^41 * (7^2)^10 * 36
= 2^(3*41) * 7^(2*10) * 36
= 2^123 * 7^20 * 36.

Therefore, the equivalent expression with only positive exponents is 2^123 * 7^20 * 36.
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